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Which of the following are true statements about a [tex]$30^\circ-60^\circ-90^\circ$[/tex] triangle? Check all that apply.

A. The hypotenuse is twice as long as the shorter leg.
B. The longer leg is twice as long as the shorter leg.
C. The longer leg is [tex]$\sqrt{3}$[/tex] times as long as the shorter leg.
D. The hypotenuse is [tex][tex]$\sqrt{3}$[/tex][/tex] times as long as the longer leg.
E. The hypotenuse is twice as long as the longer leg.
F. The hypotenuse is [tex]$\sqrt{5}$[/tex] times as long as the shorter leg.

Sagot :

Let's identify the true statements about a [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangle by analyzing its properties.

For such a triangle:
- The side opposite the [tex]\(30^\circ\)[/tex] angle is the shortest, called the shorter leg.
- The side opposite the [tex]\(60^\circ\)[/tex] angle is the longer leg.
- The side opposite the [tex]\(90^\circ\)[/tex] angle is the hypotenuse.

The key properties of a [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangle are:
1. The hypotenuse is twice the length of the shorter leg.
2. The longer leg is [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg.

Let's consider each statement one by one:

- Statement A: The hypotenuse is twice as long as the shorter leg.

This statement is true. It's a known property of [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangles.

- Statement B: The longer leg is twice as long as the shorter leg.

This statement is false. The longer leg is actually [tex]\(\sqrt{3}\)[/tex] times the length of the shorter leg, not twice.

- Statement C: The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.

This statement is true. This is another known property of [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangles.

- Statement D: The hypotenuse is [tex]\(\sqrt{3}\)[/tex] times as long as the longer leg.

This statement is false. The hypotenuse is not [tex]\(\sqrt{3}\)[/tex] times the longer leg. Instead, it is twice the length of the shorter leg. Given that the longer leg is [tex]\(\sqrt{3}\)[/tex] times the shorter leg, the hypotenuse is [tex]\(\frac{2}{\sqrt{3}}\)[/tex] times the longer leg, which simplifies to [tex]\(\frac{2\sqrt{3}}{3}\)[/tex] times the longer leg, not [tex]\(\sqrt{3}\)[/tex].

- Statement E: The hypotenuse is twice as long as the longer leg.

This statement is false. The hypotenuse is twice the length of the shorter leg, and the longer leg is [tex]\(\sqrt{3}\)[/tex] times the shorter leg. So, the hypotenuse is [tex]\(\frac{2}{\sqrt{3}}\)[/tex] times the longer leg, which simplifies to [tex]\(\frac{2\sqrt{3}}{3}\)[/tex], not twice.

- Statement F: The hypotenuse is [tex]\(\sqrt{5}\)[/tex] times as long as the shorter leg.

This statement is false. The hypotenuse is exactly twice the length of the shorter leg, not [tex]\(\sqrt{5}\)[/tex].

Therefore, the true statements about a [tex]\(30^\circ - 60^\circ - 90^\circ\)[/tex] triangle are:

A. The hypotenuse is twice as long as the shorter leg.

C. The longer leg is [tex]\(\sqrt{3}\)[/tex] times as long as the shorter leg.